A Note on Representations of the $w_\infty$ Algebra

TL;DR

This paper shows that the unitary representations of the $w_$ algebra and its truncations are equivalent to those of the Virasoro algebra, clarifying their relationship in mathematical physics.

Contribution

It explicitly demonstrates the equivalence of unitary representations between the $w_$ algebra, its truncations, and the Virasoro algebra.

Findings

Unitary representations of $w_$ algebra are the same as those of Virasoro algebra.

Truncations of $w_$ algebra also share the same unitary representations.

The result simplifies understanding of the representation theory of these algebras.

Abstract

We explicitly demonstrate that the unitary representations of the $w_\infty$ algebra and its truncations are just the unitary representations of the Virasoro algebra.